
<} }< Polynomial Time Algorithms to Approximate Permanents and Mixed Discriminants Within a Simply Exponential FactorU Alexander Barvinok Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109; e-mail: [email protected] Recei¨ed 30 April 1997; re¨ised 4 June 1998; accepted 15 July 1998 ABSTRACT: We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a nonnegative matrix and, more generally, the mixed discriminant of positive semide®nite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of OcŽ n.Ž., where n is the size of the matrix matrices and where cf0.28 for the real version, cf0.56 for the complex version, and cf0.76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting. Q 1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 29]61, 1999 Key Words: permanent; mixed discriminant; randomized algorithms; approximation algorithms 1. INTRODUCTION In this paper, we construct a family of randomized polynomial time algorithms to approximate the permanent of a nonnegative matrix. In particular, one of our algorithmsŽ. the quaternionic algorithm of Section 2.3 provides the best known * This research was partially supported by Alfred P. Sloan Research Fellowship, by NSF Grants DMS 9501129 and DMS 9734138, and by the Mathematical Sciences Research Institute, Berkeley, CA through NSF Grant DMS 9022140. Q 1999 John Wiley & Sons, Inc. CCC 1042-9832r99r010029-33 29 30 BARVINOK polynomial time approximation for the permanent of an arbitrary nonnegative matrix. Our approximation algorithms generalize naturally to mixed discriminants, quantities of independent interest. Possible extensions of our method and applica- tions of mixed discriminants to problems of combinatorial counting are discussed in the last two sections. 1.1. Permanent s = Let A Ž.aijbe an n n matrix and let S n be the symmetric group, that is the group of all permutations of the setÄ4 1, . , n . The number, n s per A ÝŁais Ži. sg is1 Sn G is called the permanent of A. We assume that A is nonnegative, that is aij 0 for all i, js1,...,n.If A is a 0-1 matrix, then per A can be interpreted as the number ¨ ¨ of perfect matchings in a bipartite graph G on 2n vertices 1,..., n and u1,...,un, ¨ s where Ž.ij, u is an edge of G if and only if a ij1. To compute the permanent of a given 0-1 matrix is a aP-complete problem and even to estimate per A seems to be dif®cult. Polynomial time algorithms for computing per A are known when A has some special structure, for example, when A has a small rankwx 5 , or when A is a 0-1 matrix and per A is smallŽ seewx 14 and Section 7.3 of wx 25. Since the exact computation is dif®cult, a natural question is how well one can approximate the permanent in polynomial time. In particular, is it true that for any e)0 there is a polynomial timeŽ. possibly randomized algorithm that approximates the permanent of a given matrix within a relative error e? In other words, does there exist a polynomial time approximation scheme? Polynomial time approxima- tion schemes are known for dense 0-1 matriceswx 15 , for ``almost all'' 0-1 matrices Žseewxwx 15, 12 , and 27. and for some special 0-1 matrices, such as those correspond- ing to lattice graphsŽ seewx 16 for a survey on approximation algorithms. However, no polynomial time approximation scheme is known for an arbitrary 0-1 matrixŽ see wx18 for the fastest known ``mildly exponential'' approximation scheme. Inwx 6 , the author suggested a polynomial time randomized algorithm, which, given an n=n nonnegative matrix A, outputs a nonnegative number a approxi- mating per A within a simply exponential in n factor. The algorithm uses random- ization, so a is a random variable. The expectation of a is per A and with high probabilityŽ. say, with probability at least 0.9 we have c n per AFaFC per A,Ž. 1.1.1 where C and c)0 are absolute constantsŽ. with cf0.28 . However, as usual, the probability 0.9 can be improved to 1ye by running the algorithm independently OŽlog ey1 . times and choosing a to be the median of the computed a s. Recently, N. Linial, A. Samorodnitsky, and A. Wigdersonwx 22 constructed a polynomial time deterministic algorithm, which achievesŽ. 1.1.1 with Cs1 and cs1ref0.37. The algorithm uses a scaling of a given nonnegative matrix to a doubly stochastic matrix. POLYNOMIAL TIME ALGORITHMS TO APPROXIMATE PERMANENTS 31 In this paper, we present a family of randomized polynomial time algorithms for approximating the permanent within a simply exponential factor. We present real, complex, and quaternionic versions of an unbiased estimator, each achieving a better degree of approximation than the previous one. Our estimators produce a number a whose expectation is the permanent and which with high probability satis®esŽ. 1.1.1 , where cf0.28 for the real algorithm, cf0.56 for the complex algorithm, and cf0.76 for the quaternionic algorithm. The last algorithm provides the best known polynomial time approximation for the permanent of an arbitrary nonnegative matrix. The algorithms have a much simpler structure and are easier to implement than the algorithm ofwx 6 . The ®rst versionŽ seewx 7. of the paper contained the real algorithm only. The complex algorithm was suggested to the author by M. Dyer and M. Jerrumwx 8 . Building on the complex version, the author constructed the quaternionic version. 1.2. Mixed Discriminant = Let Q1,...,Qn be n n real symmetric matrices and let t1,...,tn be variables. q ??? q Then detŽ.tQ11 tQnnis a homogeneous polynomial of degree n in t1,...,tn. The number, ­ n DQ,...,Q s det tQq ??? qtQ Ž.1 n ­ ??? ­ Ž11 nn . t1 tn is called the mixed discriminant of Q1,...,Qn. Sometimes the normalizing factor r wx 1 n! is usedŽ cf. 21.Ž. The mixed discriminant DQ1,...,Qn is a polynomial in the s s s entries of Q1,...,Qnkij: for Q Ž.q , k : i, j 1,...,n, k 1,...,n, we have n DQ,...,Q s sgn s sgn s qs s . 1.2.1 Ž.1 n ÝŁ Ž.Ž.1212Žk. Ž k., k Ž. s s g ks1 12, Sn The mixed discriminant can be considered as a generalization of the permanent. Indeed, fromŽ. 1.2.1 we deduce that for diagonal matrices Q1,...,Qn, where s QiidiagÄ4a 1,...,ain , we have s s DQŽ.1 ,...,Qnijper A, where A Ž.a . Mixed discriminants were introduced by A. D. Aleksandrov in his proof of the Aleksandrov]Fenchel inequality for mixed volumesŽwx 2 , see also w 21 x. The relation between the mixed discriminant and the permanent was used in the proof of the van der Waerden conjecture for permanents of doubly stochastic matricesŽ seewx 9. G It is known that DQŽ.1,...,Qn 0 provided Q1,...,Qn are positive semide®nite Žseewx 21. Just as it is natural to restrict the permanent to nonnegative matrices, it is natural to restrict the mixed discriminant to positive semide®nite matrices. Mixed discriminants generalize permanents but they also have some indepen- dent applications to computationally hard problems of combinatorial counting, some of which we describe in this paper. Suppose, for example, we are given a connected graph with nq1 vertices, whose edges are colored in n colors. Then the number of spanning trees having exactly one edge of each color can be expressed 32 BARVINOK as the mixed discriminant of some n positive semide®nite matrices, explicitly computed from the incidence matrix of the graph. Mixed discriminants play an important role in convex and integral geometryŽ seewx 21. and the problem of their ef®cient computation]approximation is not less interestingŽ but certainly less publicized. than the problem of ef®cient computation]approximation of perma- nents. Inwx 6 , the author suggested a randomized polynomial time algorithm, which, a given n positive semide®nite matrices Q1,...,Qn, computes a number , which with high probability satis®es the inequalities, n FaF ? c DQŽ.1 ,...,Qn C DQ Ž.1 ,...,Qn , where c)0 and C are absolute constantsŽ. with cf0.28 . In this paper, we construct a family of algorithmsŽ. again, real, complex, and quaternionic , which achieve cf0.28Ž. real , cf0.56 Ž complex . , and cf0.76 Ž quaternionic . The algo- rithms are natural generalizations of the permanent approximation algorithms. The real algorithmŽ. Section 3.1 can be interpreted as a ``parallelization'' of the algorithm fromwx 6 . One can note that the permanent approximation algorithm of wx22 does not obviously generalize to mixed discriminants. The paper is organized as follows. In Section 2, we describe the permanent approximation algorithms. In Section 3, we describe the mixed discriminant approximation algorithms. In Section 4, we prove some general inequalities for quadratic forms on Euclidean spaces endowed with a Gaussian measure. The results of the section constitute the core of our proofs. In Section 5, we prove a technical martingale-type result, which we use in our proofs. In Section 6, we prove the main results of the paper. In Section 7, we discuss possible extensions of our method. In particular, we discuss how to approximate hafnians and sums of subpermanents of a rectangular matrix. In Section 8, we discuss some applications of mixed discriminants to counting. 1.3. Notation Our approximation constants belong to a certain family which we de®ne here.
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